F. William Lawvere is one of the greatest category theorists of all time. He invented Lawvere theories as a category-theoretic way to describe finitary algebraic theories. He generalised Grothendieck toposes to elementary toposes, revolutionising the foundations of mathematics; in this vein, he developed the foundation ETCS. He also works on synthetic differential geometry. His motivation for all of this, believe it or not, is to better understand classical physics.
(just random items for the moment, should be expanded to a comprehensive list)
Toposes of laws of motion , transcript of a talk in Montreal, Sept. 1997 (pdf)
(on the description of differential equations in terms of synthetic differential geometry)
Outline of synthetic differential geometry , seminar notes (1998) (pdf)
Taking categories seriously, Reprints in Theory and Applications of Categories, No. 8, 2005, pp. 1–24. (pdf)
State Categories, Closed Categories, and the Existence Semi-Continuous Entropy Functions , IMA reprint 86 (pdf)
Functional Remarks on the General Concept of Chaos , IMA reprint 87 (pdf)